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Multivariate skewness and kurtosis
Journal of the Korean Data & Information Science Society 2018;29:71-81
Published online January 31, 2018
© 2018 Korean Data and Information Science Society.

Chong Sun Hong1 · Jae Hyun Sung2

12Department of Statistics, Sungkyunkwan University
Correspondence to: Professor, Department of Statistics, Sungkyunkwan University, Seoul 03063, Korea. E-mail:
Received December 12, 2017; Revised January 7, 2018; Accepted January 8, 2018.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, univariate and bivariate skewness and kurtosis are extended to multivariate statistics which could measure the degree and direction of the bias as well as the degree of thickness of the multivariate distribution tail. The k variate skewness and kurtosis are defined by dividing the functions of 2k − 1th and 2kth central moments by the (2k − 1)/2 and k squares of the determinant of the variance-covariance matrix, respectively. The skewness consists of k real numbers, and the kurtosis is expressed as a single positive real value. It was shown that the skewness and kurtosis are appropriate statistics for measuring the degree and direction of the bias as well as the thickness of the multivariate distribution function through the various trivariate normal mixtures that can be visually confirmed. Therefore, the proposed skewness and kurtosis could be used as important descriptive statistics for statistical analysis of multivariate data.
Keywords : Central moment, descriptive statistic, determinant, k variate skewness and kurtosis, mixture.